Engineering Turbulence Modelling and Experiments 2W. Rodi and F. Martelli (Editors)(!) 1993 Elsevier Science Publishers B. V. Ali rights reserved. 947

Modelling of Turbulent Dispersion in Two Phase Flow JetsR I Issaa and P J Oliveirab

ar>epartment of Mineral Reoources EngineeringImperial College of Science, Technology and Medicine, London, UK

bDepanamento de Electromecânica, Universidade da Beira Interior, 6200 Covilhã, Portugal

AbstractThe paper describes the application of a turbulence model especially developed for two-

phase flows to the pred:iction of dispersion of partic1es in co-axial confined air jets.In this model, the transport equations describing dispersed two-phase flow in an Eulerian

frame are ensemble averaged using phase-fraction weighted quantities. A number of terrosarise from the averaging process in two-phase flow in addition to those which result from asimilar averaging process in single-phase flow. The effect of these additional terms on theprediction of partic1e dispersion is evaluated by comparing calculations with existing data forthe case of two co-axial jets, one of which is particle laden.

The results show that two terms in the averaged equations are mainly responsible indetermining the computed rate of dispersion. They also show that the assumed partic1eresponse to eddy fluctuations has a marked influence on these predictions.

1. INTRODUCTION

There are a number of two-phase flow phenomena in engineering for which reasonablepredictions cannot be obtained by application of single-phase turbulence models; such modelsdo not account for the dispersion of one phase into the other by the action of the eddies.Two-phase flow models have been developed in the recent past and the present work isconcerned with the development and assessment of one such mode!.

The base model was proposed by Gosman et al (1989) and developed and implementedby Politis (1989). It is based on the two-fluid concept (Ishii 1975) in which transportequations are formulated for each of the phases. These equations are then ensemble averagedusing the phase-fraction as a weighting quantity for the averaging of alI other fluctuatingquantities; this is akin to Favre averaging in variable density flow. Such an averagingprocess possesses the distinct advantage of leading to equations with far less terms than whatis obtained from straight ensemble averaging (e.g. Elghobashi & Abou-Arab 1983). Themodel employs the eddy viscosity concept and equations for k and e are derived aloo usingphase-fraction weighting.

Correlations appear in the averaged equations which involve fluctuations of both thecontinuous and dispersed phase velocities. Such terros are modelled with the aid of theassumption that the dispersed velocity fluctuation is directly proportional to that of thecontinuous phase, the proportionality factor (Ct) being obtained from a consideration of theresponse of a single partic1e traversing an eddy.

The objective of this paper is to assess and develop the two-phase turbulence modelmentioned above. The equations involved are solved by a numerical finite-volume

948

methodology which is detailed by Oliveira (1992) and need not be described here. Themethod was applied to the problemof a two-phaseparticulatejet formed by an inner air jetladen with solid particles, confined by an externallow-velocity unladen air stream. Thenumerical results obtained after the systematic inclusion of each term of the extendedturbulence model were compared with experimental measurements of Hishida & Maeda(1991) therebyenablingan assessmentof the moreimportantfactors in the turbulencemodel.

This assessmentshows that someof the extra terms in the turbulencemodel are requiredin order to predict dispersion of the particle phase; however, additional refinement of themodel is still neededfor predictingthis dispersionaccurately.

2. EQUATIONS AND TURBULENCE MODELLING

The extension of the single-phase k-e turbulence model to two-phase flows is nowpresented. This is done by introducing into the equations of motion, turbulence kineticenergy (k) and dissipation (e), the additional terms resulting from correlations of volumefractionand velocities. This followsthe workofGosman et al (1989)and Politis (1989).

2.1. The a-weighted, ensemble-average equations of motionThese are based on the Eulerian treatment of both phases, following the two-fluid model

(Ishii 1975). The continuity and momentum equation obtained after applying a doubleaveraging procedure (volume-average followed by ensemble-average) to the usual single-phase equations can be written as (Oliveira 1992):

(1)

p~ i ~k~k + V.~k~k~k) =-~kVp+ ~kV.~k+ V'~k~~+ Pk~kg + Fl\ . (2)

In these equations the subscript k denotes the phases (c for the continuous and d for the

dispersed) and p, a, p, u and t are the density, volume-fraction, pressure, velocity vectorand stress tensor. The turbulent stresses are denoted with a superscript t. The interphasemomentum exchange is represented by FD' resulting from the action at the interface of the

~ressure forces (Fp) and viscous stresses (F't)' The sum of F't with part of Fp can be

identified as the usual drag force, whereas the remaining part of Fp contributes to the virtualmass and inviscid lift forces, which will not be considered here. The reason for neglectingthese terms is based on an order-of-magnitude analysis for the case of particle laden air jets

where piPc - HP. In equations (1) and (2) the overbar is used, as usual, to denote

ensemble-averaging. The symbol - is used to denote a-weighted averaging, which is similar

to Favre averaging used in variable density single-phase flows. The definition of a-weightedaveraging is:

111= allll a (3)

where 111represents any phase averaged quantity (i.e. one obtained after the first volume-average operation) and which may be split into a mean plus a fluctuating value, as:

949

111 = 111+ 111" = j; + 111'

The expression for the turbulent stress in the momentum equation (2) is:

(4)

2.2. Main modelling assumptionsTo solve the momentum equation for each phase it is necessary to define the correlations

appearing in those equations. The turbulent stress (eqn. 4) for each phase is modelledfollowing the Boussinesq approximation:

(5)

where Ôis the identity tensor. For the continuous phase, the turbulence kinetic energy kc will

be obtained from its own transport equation and the turbulent viscosity ~cl is given by the k-Emodel, as explained in section 2.5. On the other hand, the dispersed phase turbulentviscosity and kinetic energy need to be specified as functions of the respective continuousphase values. To do this, and to develop alI the correlations, two main model assumptionsare required.

The first is the gradient diffusion for the transport of volume fraction by velocityfluctuations:

In these equations TIis the turbulent diffusivity of a which will be obtained from Tlc=vl/(Ja'

with the "Schmidt" number here taken as unity, (Ja = 1.The second main model assumption (introduced by Gosman et aI 1989) links the

instantaneous velocity fluctuations of one phase to the velocity fluctuations of the other. Thisis a key point in the modelling and it is expressed as:

(8)

where the turbulence correlation function C1is given by:

c = l-exp(-t/t).I E P (9)

Expression (9) is derived from the integration of the Lagrangian equation of motion of aparticle:

a u' = -TI Va , (6)c c c c

- -a dU = - TldVa d . (7)

950

I I ,

dUd (uc - Ud)

dt = tp

subject to the initial condition t = O,Ud'= O,and where the relaxation time and the fluidfluctuating velocity are assumed constant. The two time scales in eqn. (9) are the eddylifetime (tE)and the particle relaxation time (~), which take the forros:

tE = 0.4( ;) (10)

tp = ((1-ad)/Ao) (I+CM::)

wherethedrag factorAo is givenin the nextsectionand the virtualmasscoefficientCMtakesthe value 0.5. Note that the virtual mass effect is important in eqn. (11) regardless ofwhetherit is includedin the averagedmomentumeqn. (2) or noto

Typical values of relaxation time is 28 ms for the present particulate flow (dp =64 Jl, ur~

=0.28 m/s, Rep:: 1).It can also be shown that the two diffusivities in eqns. (6) and (7) are related by:

Tld = CtTlc'

(11)

(12)

For responsive particles Ct = 1, and the two diffusivities are identical.

2.3 Modelling the drag forceDrag is modelledassumingthat thedispersedphase is composedof many small spherical

particles which do not interact with each other -this interaction is absorbed imo the dispersedphase stress tensor. This is true for the low volume-fractions encountered in most particle

ladenjets (a == ad - 10-4). The drag force is first linearised and modelIed as:

Foc = Fo (üd - üc) (13)

where

(14)

In these expressions, ur is the relative or slip velocity (ur = IIud - Uc 11), ~ is the particlediameter, and the drag coefficient Co is given as a function of the particle Reynolds number

(Rep =urCYVc' Vc =Jljpc) by an empirical relationship such as:

Co = R24 (1 + 0.15Reo.687 ).e pp

(15)

Unlike the work of Gosman et al (1989) and others such as McTigue (1983) the drag term ishere modelIed after, and not before, the second (ensemble) averaging operation; the result

951

however is the same. Now, sinee ultimately the veloeities solved for are the a-weightedones it becomesneeessaryto transforroü intou.This ean be done by using the usual Favre-averagerelations(seeeqns. (3) (6) and (7», to obtain:

(16)

where it has been assumed that the two diffusivities Tldand Tlcare equal. It ean be seen thatthe drag is eomposed of the usual mean drag plus a eontribution proportional to the void-fraction gradient whieh arises from turbulent fluetuations of a and u. The eontribution of the"turbulent drag" ean be quite important, mainly in the radial direction where volume-fraetiongradients are high and the mean drag is small.

2.4. Modelling the dispersed phase turbulent stress and kinetic energyThe expression for the turbulenee kinetic energy of the dispersed phase is:

(17)ak

whieh is somewhat similar to the expressionfor tdt (eqn. 4). The simplifiedexpressionsto

be used for kdand ti turn out to be:

(18)

Pd -tC - t

k Pc c

where Ck =C? The turbulent kinematic viseosity of the dispersed phase is obtained byeomparing eqn. (19) with the Boussinesq stress model (eqn. 5) to obtain:

t tVd =Cv

v c

(19)

(20)

where again Cy =C?-

2.5. The k and E equations and the modelling of the additional termsThe a-weighted equationsfor the transpon of turbuleneekinetieenergy (k) and its rate of

dissipation(E),for the eontinuousphase, arewritten as:

{ )

ta - - - - - - Jl - - - k

P :.. a k + V.a u k = V.(a ~ V k) + a (G -P E ) + SdU( c c. c c c c c:r c c c ck

(21)

{a - - - - -

)

- Jlt - - E -P :.. a E + V.a u E = V.(a ~VE ) + a ~C IG - C2P E ) + SEd .

U(cc ccc Cc:r c c- ccE kc

(22)

952

These equations are a standard generalisation of the single-phase k-e model (Jones andLaunder 1972) applied to the continuous phase, except for the additional tenns Sd whichaccount for the interaction between dispersed and continuous phase turbulence. Theturbulentor eddyviscosityand the generationofk arecomputedfrom:

í?- C -=-- 11 - ,

ee

tV

e (23)

t - - ..:rG = Jl V u . (V u + V u ),e e. e e

and the constants used in the present work are the standard ones (C1 = 1.44,C2 = 1.92,CI1

= 0.09, <1k= 1.0, <1e= 1.22).

The additional drag-related source tenns in the k and e equations arise fiom the timeaverage of the inner product between the instantaneous drag force and the fluctuatingcontinuous-phase velocity. This term when expanded and then modelled using the precedingassumptions becomes:

k

~

T\e - - - -)

Sd =-F =-=(ud - ue)' V<Xd+ 2ke (1- Ct)<Xe<Xd

The main contribution for Sk is given by the last term in eqn. (25), which constitutes a sinkof turbulence energy because Ct :s;1. It will induce a dissipation equal to the tenn divided by

the turbulence time scale (k/e). Hence the additional source in the e-equation is modelled as:

(24)

(25)

(26)

where the model coefficient <; is taken as unity.

2.6. Alternative models for CtInitial computationsby Issa and Oliveira (1991)with the model ofGosman et al (1989)

for the case of particleladenjets treatedherein showedthat the rate of dispersionof particleswas grossly underpredictedby the model. A subsequentstudyrevealed that two terms in themodelplay the dominantrole in determiningthedispersionrate; theseare:

2 - - t --'3 V (CkPd<Xdkd) and - (FD ve/ (Ja<Xe<Xd)V<Xd

both of which arise in the dispersed phase momentum equations. The first originates fiommodelling the turbulent stresses (eqn. (5» and the second results fiom considering thefIuctuation of the drag term (eqn. (16». The magnitudes of these terms affect the transversedispersed phase velocities and thereby the rate of dispersion of that phase.

Both of the above terms depend on the value of the particle response coefficient Ct (eqns.(18) and (20». This suggests that the dispersion rate is closely linked to the value of Ct asindeed was discovered by numerical experimentation with different models for calculating C!,

953

Those niOOwere amongst the models found in the literature that are listOObelow:

[ ]-1/2

C~ = 1 + 0.45 u:{ ~ k) / C~ (Mostafa & Mongia 1987)

Furthermore, many workers use values for <; and Cy in eqns. (18) and (20) that aredifferent from Ct 2 as proposed by Gosman et aI (1989). For example, Melville & Bray1

(1979) use Ct2for both <; and Cy' Chen & Wood (1985) take Ct2for Cy whereas Mostafaand Mongia (1987) use Ct2for <;. These alternatives have also been nied in the presentwork.

3. RESULTS

The geometry to which the model is appliOOconsists of two vertical co-axial pipes (Fig.1): the inner pipe carries a mixture of glass sphere particles and air; the outer pipe is used toconfine the jet and carries a lower velocity air-stream. The inner pipe diameter is D = 13 mm(R =D(2) and the diameter of the outer pipe is D2 =60 mm. For the numerical simulation thelength of the domain '(along x) was taken as 45 times the inner diameter (D).

The physical properties are: Pc = 1.18 Kglm3, Jlc = 1.8 10-5 KgI(m.s), Pd = 2590Kg/m3. The average particle diameter is 64.4 JlII1.

The inlet profiles ofaxial and radial velocity, and of turbulence kinetic energy are givenby the experimenters (Hishida & MaOOa1991). The centre-line mean values at inlet are: Uco

=29 m/s, Udo = 23 m/s, and ao = 2.5 10-4. The air velocity in the outer pipe at inlet (U2 inFig. 1) is almost constant at 15.6 m/s as could be observed from the measurOOvelocityprofile.

The numericalmesh overlayingthe 2-D (axial-radial)physicaldomainconsistOOof 50 x48 non-uniformlydisnibutOOinternalcells. The overall dimensionsof the solutiondomainare 600 mm x 30 mm (axial and radial directions) and is bounded by an inlet (x=O),outlet(x=O.6m), axis of symmetry(y=O)and wall (y=O.03m). The mesh is more concentratOOinthe region betweenthe axis and the line y=6.5 mm (y=R),which is the radius of the innerjetat inlet, and thenexpandsin the directionof theouter wall,whereit contractsagain.

Fig. 2 shows the radial profiles of the particle flux (which is a measure of the phasefraction) normalisOOby its centre line value at two axial stations. Calculations with the

c = 1-exp(-t / t ) (Gosmanet al)lt e p

C = [1+ tp/ ter(Faeth 1987)

[ rCt4 = 1+0.45u;/ ( k)(Csanady1963,Picartetal1986)

C =[1 + t / (Ct t) r (Simonin 1991)Is p 4 e

954

standard model of Gosman et al (with Cy =Ct2 as defined in eqn. (20» ean be clearly seen tounderprediet the dispersion effect exhibited by the data. An attempt to enhance the dispersivemechanism by taking Cy =1gives someimprovementbut not quite what is sought This eanbe explained by examining the quantities responsible for the particle dispersion: themagnitudes of the fluetuations of the dispersed phase. Fig. 3 shows the plOfiles of the rmsfluctuations of the radial and axial velocities respectively at the second measuring station.The first thing to note is the anisotropy of the turbulenee whieh eould not be eaptured by the

present k and E model. However, this is on1y part of the reason why the predietions with the

standard model (with C; =Ct12 as defined in eqn. (18» give too low a spread rate. FlOmFig. 3 it is apparent that the fluetuations whieh are responsible for the dispersive effect aregrossly underpredieted. It takes the use of Ct2 and C... for C; (see section 2.6) to increase

the eomputed fluetuations up to the measured levels of vd' and ud"In Fig. 4, the plOfiles for the normalised particle flux at the two axial stations are

displayed for different formulations of C;. Comparison with the data shows that like Ctl as

suggested by Gosman et al, C~ also gives too low a dispersion rate. On the other hand C...(see section 2.6) yields too high a value. The sensitivity of the predietions to the value of C;.

ean also be gleaned from the predietions when C; is ehosen to be 0.5 Ct4; the dispersion rateis now much lower than what is measured suggesting that further refinement of the model forealeulating Ct is essential in arder to prediet the real behaviour bener.

4. CONCLUSIONS

The paper presents developments and applieation of a turbulenee model for two-phaseflows. The model is applied to the predietion of a particle-laden jet flow whieh is aeonvenient plOblem for model assessment sinee the dispersion of the partieles is mainly dueto turbulence effects. This modelleads to additional terms in the equations as compared with

the standard k-e model applied to the eontinuous phase. Sueh terms were introdueedsystematieally in the equations and the resulting predietions were analysed and comparedwith data. The main eonclusions from this study are:

1. The standard model of Gosman et aI (1989) prediets dispersion of the particle-phase;however the dispersion is under-predieted, as revealed by a eomparison with measuredparticle-flux profiles.

2. The main terms plOmoting dispersion are the turbulent drag and the -2/3VC;ak term inthe dispersed-phase radial momentum equation; in this last term, Ck:should be higher

than the function Ct12as mentioned earlier and smaller than 1.

3. The dispersed-phase eddy-viscosityobtained by sening vdt=Ct/vct appears to be too

small; with sueh small vdt(=O) the results become very sensitive to the imposed nidial

velocity for both phases at inlet. With higher vi (either vi =vct or vi = Ct4 Vct), the

results are not sensitive to the given inlet radial-velocities.

4. If the C(funetion used in the term -2I3VCk:ak is too high (e.g. C; = Ct4)' an overshoot

of the distribution of a along the axis oceurs close to inlet (x/D ~ 5). This overshoot is

955

more aceentuated if vi is small. However, even for medium or high Vdt the overshoot is

present, although less aeeentuated. Use of C;=Ctl or Ck=Ct2 (related to inertia effects

only) under-prediet the dispersion; use of C;=Ct4 (related with erossing-trajeetorieseffeet) yields over-predietions. This suggests use of a CCfunetion for Ck whieh takesinto account both effects (inertia and crossing-trajectories).

5. The predietions of particle-dispersion are mueh improved with the modifieations to the

Ccfunetions mentioned above; agreement with the particle-flux data (and also with a) isstill not perfeet but it is similar to other authors (in Sommerfeld & Wennerberg 1991).

Further improvement eould be obtained by using a Sehmidt number (Ja smaller than 1 inthe turbulenee drag term (as Simonin 1991).

7. REFERENCES

1 Chen, c.P. and Wood, P.E. ~ A turbulenee closure model for dilute gas-partieleflows. CanoJ. Chem. Eng. 63, pp. 349-360.

2 Csanady,G.T. ~ Turbulent diffusion of heavy partic1esin the atmosphere. J. Arm.Sei. 20, pp. 201-208.

3 EIghobashi, S.E. and Abou-Arab, T.W. .l2.81 A two-equation turbulenee model fortwo-phase flows. Phys. F/uids 26, pp. 931-938.

4 Faeth, G.M. 12ll Mixing, transport and eombustion in sprays. Prog. EnergyCombust. Sei. 13, pp. 293-345.

5 Gosman, A.D., Issa, R.I., Lekakou, C., Looney, M.K. & Politis, S. l2..8..2Multidimensionalmodellingof turbulent two-phase flows in stirred vessels. In AiChEannua/meeting, 5-10 Nov.

6 Hishida, K. and Maeda, M. .l22.1 Turbulenee eharaeteristies of gas-solids two-phaseeonfined jet (Effeet of partiele density). In Proc. 5th Workshop on Two Phase F/owPredictions, Erlangen,March 19-22, 1990,pp. 3-14.

7 Ishii, M. .1lli Thenno-F/uidDynamic Theory qfTwo-Phase F/ow. Eyrolles, Paris.8 Issa, R.I. and Oliveira,P.I. .l22.l Method for predietionof partieulatejets. In Proc. 5th

Workshopon Two Phase F/ow Predictions,Erlangen,March 19-22, 1990,pp.39 & 41.9 Jones, W.P. and Launder, B.E. .l2li The predietion of laminarisation with a two-

equation model of turbulenee. Int. J. Heat Mass Transf. 15, p. 301.10 MeTigue, D.F. !2ll Mixture theory of turbulent diffusion of heavy particles. In

Theory of Dispersed Mu/tiphaseF/ow, Ed. R. Meyer, AeademiePress, pp. 227-250.11 Melville, W.K. and Bray, K.N.C. l212 A model of the two-phase turbulentjet. Int.

J. Heat Mass Transf 22, pp. 647-656.12 Mostafa, A.A. and Mongia, H.C. 1m On the modelling of turbulent evaporating

sprays:Eulerianversus Lagrangianapproach.Int J HeatMass Transf30 pp 2583-2593.13 Oliveria, P.J. 1992 Computer modelling of multidimensional multiphase flow and

applieation to T-junetions. PhD Thesis, Apri192,Univ. ofLondon.14 Pieart, A., Berlemont,A. and Gouesbet,G. .lilllli Modellingand Predieting turbulenee

fields and the dispersion of diserete particles transported by turbulent flows. Int. J.Mu/tiphase F/ow 12, pp. 237-261. .

15 Politis, S. .l2.8.2Predietion of two-phase solid-liquid turbulent flow in stirred vessels.PhD Thesis, ImperialCollege,UniversityofLondon.

16 Simonin, O. .l22.1 An Eulerian approaeh for turbulent two-phase flows loaded withdiserete partieles: eode deseription. In Proc. 5th Workshop on Two Phase F/owPredictions, Erlangen,March 19-22, 1990,p.40 and pp 156-166.

17 Sommerfeld, M. and Wennerberg,D. (Eds.) 1.2.2.LProc.5th Workshopon Two PhaseF/ow Predictions. Erlangen, Mareh 19-22, 1990. Fersehungszentrum Jülieh Cmbh,KFA.

956

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~ 0.4

',--

,

:'"u,, ', ', ," '

',I:'I,

, b 'i'-,I .

Lf/2

1REGlONI

.1

I

I

".J

REGIONI

I

I

Fig.l Experimental FIow Configuration

1.0

.g 0.6q :"-

~ 0.4

X/D=10

0.8 - C".=1 2--_n C".=C ~

doto

0.2

0.0o 2

Y/R3 4

Fig.2

1.0

0.8 - C".=1 2--n- C".=C~

do to

0.2

0.0

o 2

Y/R

43

Radial Profiles of the Nonnalised ParticIe FIuxat Two Axial Stations with Different <;.

0.5X/D=20- pred. ~

pred. r:0.4 ... .

/"""',

(jJ 0.3"C

'-' 0.2

., ." -- ,...'" " - ~--'" '... ~... "',...- -'" ~ "> 0.1

0.0o 2 3

Y/R

4

(jJ" 1 0C .

2.0X/D=20

- pred. k_uuu pred. r:

1.5 ..... .

'"~" 0.4~

0.0o 2 3

Y/R

Fig.3 Radial Profiles of the RMS FluctuatingDispersed Phase Velocities

1.0 X/D=10

0.8 -c,=c..- - - - C,=C"

nnm C,=0.5Ç.dolo0.6'"

~" 0.4~

0.0

o 2

Y/R

3 4

Fig.4

1.0 X/D=20

0.8 -C,=C..C,=C"

_Um- C,=0.5Ç.dolo0.6

0.2

0.0

o 2

Y/R

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Radial Prof11esof the Nonnalised Particle Fluxat Two Axial Stations with Different C1c

957

4